Weighted composition operators between vector-valued Bloch-type spaces

Let X " role="presentation"> X X X and Y " role="presentation"> Y Y Y be complex Banach spaces and D " role="presentation"> D \mathbb{D} be the open unit disc in the complex plane C " role="presentation"> ℂ C \mathbb{C} . Let φ " role="presentation"> φ φ \varphi be an analytic self-map of D " role="presentation"> D \mathbb{D} and ψ " role="presentation"> ψ ψ \psi be an analytic operator-valued function from D " role="presentation"> D \mathbb{D} into the space of all bounded linear operators from X " role="presentation"> X X X to Y . " role="presentation"> Y . Y . Y. The weighted composition operator W ψ , φ : H → H ( D , Y ) " role="presentation"> W ψ , φ : → ( , Y ) W ψ , φ : H → H ( D , Y ) W_{\psi,\varphi}:\mathcal{H} \rightarrow\mathcal{H}(\mathbb{D}, Y) is defined by W ψ , φ ( f ) ( z ) = ψ ( z ) ( f ( φ ( z ) ) ) , ( z ∈ D , f ∈ H ) , " role="presentation"> W ψ , φ ( f ) ( z ) = ψ ( z ) ( f ( φ ( z ) ) ) , ( z ∈ , f ∈ ) , W ψ , φ ( f ) ( z ) = ψ ( z ) ( f ( φ ( z ) ) ) , ( z ∈ D , f ∈ H ) , W_{\psi,\varphi}( f)(z)=\psi(z)(f(\varphi(z))), \quad \quad (z\in\mathbb{D}, f\in \mathcal{H}), where H " role="presentation"> H \mathcal{H} is the space of all analytic X " role="presentation"> X X X -valued functions on D " role="presentation"> D \mathbb{D} . In this paper we provide necessary and sufficient conditions for the boundedness and compactness of weighted composition operators W ψ , φ " role="presentation"> W ψ , φ W ψ , φ W_{\psi,\varphi} between vector-valued Bloch-type spaces B α ( X ) " role="presentation"> α ( X ) B α ( X ) \mathcal{B}_{\alpha}\left( X\right) and B β ( Y ) " role="presentation"> β ( Y ) B β ( Y ) \mathcal{B}_{\beta}\left( Y\right) for α , β > 0 " role="presentation"> α , β > 0 α , β > 0 \alpha, \beta >0 in terms of ψ , φ " role="presentation"> ψ , φ ψ , φ \psi,\varphi , their derivatives, and the n " role="presentation"> n n n th power φ n " role="presentation"> φ n φ n \varphi^n of φ " role="presentation"> φ φ \varphi .